In the result, \({d_I^1}\)  stands for the 1-presentation distance (presentation-distance-def), which is the \(p=1\)  case of the \(p\) -presentation distance, recently introduced by Bjerkevik and Lesnick [1]}.
Note that the signed 1-Wasserstein distance does satisfy the triangle inequality (properties-signed-wass)—justifying its name—and can be used to compare multiparameter persistence modules directly at the level of Hilbert functions.
Indeed, in hilbert-stability-remark and hilbert-stability-corollary we explain how wasserstein-stability can be interpreted as a stability result for Hilbert functions.
The proof of wasserstein-stability, given in stability-hilbert-decompositions, relies on key properties of the \(p\) -presentation distance established by Bjerkevik and Lesnick in [1]}.
